Different vacua in 2HDM

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by  I. F. Ginzburg

Different vacua in 2HDM
I. F. Ginzburg, K.A. Kanishev
Sobolev Institute of Mathematics and Novosibirsk State University,
Novosibirsk, Russia
We discuss the extrema of the Two Higgs Doublet Model with different physical properties. We
havefoundnecessaryandsufficientconditionsforrealizationoftheextremawithdifferentproperties
asthevacuumstateofthemodel. Wefoundexplicitequationsforextremumenergiesviaparameters
ofpotentialifithasexplicitlyCPconservingform. Theseequationsallowtopickoutextremumwith
lowerenergy–vacuumstateandtolookforchangeofextrema(phasetransitions)withthevariation
of parameters of potential. Our goal is to find general picture here to apply it for description of
early Universe.
8
0
0 I. INTRODUCTION. MOTIVATION
2

n The Two Higgs Doublet Model (2HDM) presents
a the simplest extension of minimal scheme of Elec-
J
troweak Symmetry Breaking (EWSB) allowing to in- ⇒
4 clude naturally observed CP violation and Flavour
1
Changing Neutral Currents [1]. Natural approach

in its description is to derive its parameters based
]
h from modern data and future observations at collid- t≈ 0 t > tc φ
p ers. This very approach with properties of vacuum,
- FIG. 1: Evolution of Gibbs potential in minimal SM
fixed by observations, is developed in many papers
p
e (seee.g. [2]-[4]). Suchapproachcontainsthedangerthat
h the discussed set of parameters allow another minimum
[ of potential which is deeper than the discussed one, in During the inflatory expansion, the Universe becomes

this case obtained results correspond not real but false colder, and at some temperature the Gibbs potential
3
v vacuum. transforms effectively into the well known form of the
Higgs model with φ = 0 – we obtain our world with
4 The Models like 2HDM can have different vacuum
h i 6
6 states (minima of potential) with various physical prop- massive particles, etc. (EWSB) (see Fig. 1). This phase
6 transitiondetermines the fate ofthe Universeafter infla-
erties. Another approach appears – to study different
3 tion.
vacua in order to confine field of possible values of pa-
.
4 rameters of Lagrangianallowing description of data (see Weseein2HDMmanypossiblevacuumstatesdepend-
0 e.g. [5]–[8]). ingoninterrelationoftheparametersofthepotential. In
7 In this paper we consider different possible vacuum the Gibbs potential the temperature dependent addition
:0 states in 2HDM with two goals. tothemasstermhasformT2aij(φ†iφj)/2[9]. Withthese
v 1). We like to have a complete set of necessary and terms the interrelation mentioned above changes during
Xi sufficient conditions for realization of the extrema with cooling of Universe, this leads to the change of phase
different properties as the vacuum state of the model. state of Universe. The sequence of phase states of Uni-
r
a These conditions must allow to check whether the dis- verse during its early history, transitions among vacuum
cussed minimum of potential is global one (vacuum) or states with different properties can influence for current
not. state of Universe [9].
2). Theobtainedresultsmustallowtostudy(infuture For these goals we consider all possible extremum
•
works) what can happen at variations of parameters of states of the Higgs potential and after that investigate
Lagrangian, related to the evolution of earlier Universe, whichoftheseextremacanbethevacuumstate–aglobal
as it was proposed in [9]. Let us describe this idea in minimum of potential. So, in this paper we discuss all
more details. types of extremum states in 2HDM and determine con-
At first moments after Big Bang the temperature of ditions when one ofthem is vacuum state. The obtained
the Universe T was very high, in this stage vacuum ex- explicitformofdependence ofdifferentvacuumstateen-
pectation values of Higgs fields are givenby minimum of ergies on parameters of Lagrangian seems to be an im-
the Gibbs potential V . The latter is a sum ofthe Higgs portant result on this way.
G
potentialV(φ)andthetermaT2φ2. Itcorrespondstothe In sect. II we describe the Lagrangian of the model
Higgs model with parameters varying in time. At large andits generalproperties. Sect. III is devoted to the de-
T potential has EW symmetric minimum at φ = 0. scription of different extrema of the potential and their
h i
Thisstagedescribesthewidelydiscussedphenomenonof firstclassification. Inshortsect.IVwediscussconditions
inflation. when the electroweak symmetry point φ1 = φ2 = 0
h i h i
2
can be (local) maximum or minimum of the potential.
In the sect. V we study the most exotic type of the ex-
tremum–chargedextremum,whichisnotrealizedinour V =V0 V2(xi)+V4(xi);
−
world–inthisextremumtheinteractionofgaugebosons V2(xi)=Mixi
1 ≡
wtointhbfeecrommioesnsmwasilslivneo,teptcr.es[e7r].veHtohweeevleerctitriicscnhoatrgime,pprohbo-- ≡ 2 m211x1+m222x2+ m212x3+h.c. ,
able thatthis state wasthe vacuumstate insomeperiod (cid:2) V4(xi)=Λijx(cid:0)ixj/2≡ (cid:1)(cid:3) (3)
aofft”ernoBrimgaBl”annge.utTrahlenexwtreemcoamientsoectth.eVgIe.neTrhaleidrissctuusdsyioins ≡ λ1x21+2λ2x22+λ3x1x2+λ4x3x†3+
cseornvtiinnguepdoftoerntthiaeli(mwpitohrtaanlltrceaaslecoofetffihecieexnptsli)ci–tlyseCctP. cVoInI-. + λ5x23+λ6x1x3+λ7x2x3+h.c. .
2
We discuss in detail uniquely defined doubly generate (cid:20) (cid:21)
spontaneouslyCPviolatingextrema,sect.VIIAandCP Here i, j = 1, 2, 3, 3∗, Λij = Λji. Besides, λ1−4 and
preservingextremainsect.VIIB. Thenwediscussinter- m2ii are real while λ5−7 and m212 are generally complex.
relationamongdifferentextrema,sect.VIII. Inappendix The field independent term V0 is added for convenience
A we develop a special toy model, for which all the cal- in future, we omit this term in many equations below.
culations can be done easily. This model gives a simple The reparametrization and rephasing sym-
illustration of many general statements in the main text me•try. Our model contains two fields with identical
and provides an answer to problems of realizability of quantum numbers. Therefore, the pure Higgs sector can
some situations. In sect. IX and X we summarize the be described both in terms of fields φ (k = 1,2), used
k
results obtained and briefly discuss possible applications in Lagrangian (3), and in terms of fields φ′ obtained
k
for the history of Universe. from φ by a generalized rotation. The corresponding
k
reparametrizationsymmetry was studied in [4, 6, 10].
In fact, in the description of reality we deal usually
II. LAGRANGIAN
with the Yukawa sector, where right hand isosinglet
fermion fields of each type are coupled with only one
The spontaneous electroweak symmetry breaking via basic field φ1 or φ2 (Model II, like MSSM, or Model I,
the Higgs mechanism is described by the Lagrangian1 see [2]). This property becomes hidden at the general
reparametrization transformation. The efficient form
= SM + + with =T V ,
L Lgf LH LY LH − of the potential is that in which above property of
φ+ (1) Yukawa interaction is explicit.
φ = i .
i φ0 This efficient form of potential retains one degree of
(cid:18) i(cid:19) freedom–theindependentphasetransformationoffields
Here SM describes the SU(2) U(1) Standard Model φi φieiρi and corresponding phase transformation of
interaLctgifon of gauge bosons and×fermions, describes par→ametersofpotentialareallowed–rephasing (RPh)
Y
the Yukawa interactions of fermions with HLiggs scalars transformation. RPh symmetry group is the sub-
and is the Higgs scalar Lagrangian; T is the Higgs group of reparametrization symmetry group. For La-
H
kinetLic term and V is the Higgs potential (3). In this grangianitisone-parametricgroupwithsingleparameter
paper we won’t consider effects related to possible non- ρ1 ρ2. Below we haveinmind mentionedefficient form
−
diagonal terms in T (see preliminary discussion in [4]). of potential and RPh freedom for it.
The Two Higgs Doublet Model is the simplest exten- The potential with explicit CP conservation is that
sion of the minimal SM. It contains two scalar weak with all real λi, m212 (or with parameters which can be
isodoublets φ1 and φ2 with identical hypercharge. In transformed to real ones by single RPh transformation).
particular,it is realizedin MSSM. To describe Higgs po- The results for the most general Lagrangian (pre-
•
tential in short form, it is useful to introduce isoscalar sented below) often have very complex form. Main fea-
combinations of the field operators turesofphysicalpictureareseeninmoresimplepotential
with softly broken Z2 symmetry2 , in which λ6 =λ7 =0.
x1 =φ†1φ1, x2 =φ†2φ2, Wewilldiscussmanyresultsusingfinalequationsforthis
(2) verypotential,often–intheexplicitCPconservingcase.
x3 =φ†1φ2, x3∗ x†3 =φ†2φ1. Positivity constraints. To have a stable vacuum,
≡
•
thepotentialmustbepositiveatlargequasi–classicalval-
The most general renormalizable Higgs potential is the
sumoftheoperator V2ofdimension2andtheoperator
−
V4 of dimension 4:
2 For the most general Higgs potential loop corrections with λ6, λ7
generally mix φ1 and φ2 fields even at small distances, it results
in breaking of the mentioned Model II or Model I form of Yukawa
1 Notations and main definitions follow [4], we use some equations interaction. This breaking is absent for the potential with softly
from[9]. broken Z2 symmetry,inwhichthekinetictermhasdiagonalform.
3
ues of fields φ (positivity constraints) for an arbitrary Onecanconsidereachpairoftheseequationsasasys-
k
| |
direction in the (φ1,φ2) plane. It means that upon re- tem for calculation of quantities Ti via yi. The deter-
placement of the operators xi with some numbers zi minant of these systems are precisely Z = y1y2 y3∗y3.
−
Therefore, it is natural to distinguish two types of ex-
V4(zi)>0 if z1, z2 >0, z1z2−z3z3∗ ≥0. (4a) trema, with Z =0 (”chargedextrema” with Ti =0) and
6
This condition limits possible values of λi. For the po- with Z =0 (”neutral extrema” with Ti =0).
6
tential with softly broken Z2 symmetry (λ6 = λ7 = 0) For each EWSB extremum one can choose the z
such limitations have form (see e.g. [11, 12]) • 0
axis in the weak isospin space so that hφ1i= v1 with
λ1 >0, λ2 >0, √λ1λ2+λ3 >0, (4b) real v1 > 0 (choose ”neutral direction”). In(cid:18)such(cid:19)basis
√λ1λ2+λ3+λ4−|λ5|>0. chφh2oiicheasthgeenmeorsatllgyeanneraalrbeiltercatrryowfoearmk.syTmhmenet,rayfvteiorlatthinigs
solution of (5) can be written in a form with realv1 and
III. EXTREMA OF POTENTIAL complex v2:
1 0 1 u
The extrema of the potential define the values hφ1,2i hφ1i= √2 v1 , hφ2i= √2 v2 (9)
of the fields φ1,2 via equations: (cid:18) (cid:19) (cid:18) (cid:19)
∂V/∂φ =0, ∂V/∂φ† =0. (5) with v1 =|v1|, v2 =|v2|eiξ,
i|φi=hφii i|φi=hφii
without loss of generality we can consider only real pos-
Theseequationshavetheelectroweaksymmetryconserv-
itive u.
ing(EWc)solution φ =0andtheelectroweaksymme-
i
h i Atu=0wehaveZ >0–chargedextremum,atu=0
try breaking (EWSB) solutions. Here and below nota- 6
we have Z =0 – neutral extremum.
tion F mean numerical value of the operator F at ex-
h i Mass matrix at each extremum is given by the
tremum. Ingeneral,therearemanyEWSBextrema. We •
decomposition of the fields φ near the extremum point.
label these extrema by an additional subscript, if neces- i
Sincethepositionofanextremumpointrelativetoorigin
sary; e.g. F means value of F in N-th extremum.
N
h i h i ofcoordinatesselectssomedirectioninthe isospace,this
We consider also the values y of operators x at the
i i
matrixhasdifferentelements fordifferentcomponentsof
extremum points. In the discussed tree approximation
fields
(mean field in the statistical physics) we have
∂2V
yi,N ≡hxiiN =hφai†NhφbiN for xi =φ†aφb. Mij,αβ = ∂φβ∂φ∗α ,(i, j =1, 2, α,β =+, 0). (10a)
i j
In each extremum point these values obey inequalities
following from definition and Cauchy inequality, written Direct differentiation gives
for important auxiliary quantity Z:
y1 >0, y2 >0, Z =y1y2−y3∗y3 ≥0. (6) +MΛ1∗113,αφβ2β=†T1φδ1ααβ ++ΛΛ3111hφφ11ββi††hφφ21αiα++
h i h i h i h i
Classification of EWSB extrema. Itis usefulto +Λ33∗ φ2β † φ2α ,
• h i h i
define quantities
M12,αβ =T3δαβ +Λ12 φ2β † φ1α +
Ta ≡h∂VT/∂1,x2aia=reΛarieyail−, MaT3∗(a==T13∗,.2, 3, 3∗), (7) +Λ13∗hφ1β+iΛ†h3φ31∗αhφi1+βiΛ†3h2φhh2φα2iβ,ii†hhφ2αii+ (10b)
In these terms eq. (5) have form M21,αβ =T3∗δαβ +Λ12hφ2βihφ1αi†+
+Λ13∗ φ1β φ1α †+Λ3∗2 φ2β φ2α †+
h∂V/∂φ†1i= hφ1iT1+hφ2iT3 =0, h +ihΛ33∗ihφ1βihφ2αhi†,ih i
hh∂∂VV//∂∂φφ1†2ii== hφhφ12i†iTT12++hhφφ21iiT†T3∗3∗==00,, (8) +MΛ2232h,αφβ1β=i+†ΛThφ2323δα∗αiβφ++1βΛΛ3†2∗22φhh1φφα22ββ.ii††hhφφ21ααii++
h i h i
∂V/∂φ2 = φ2 †T2+ φ1 †T3 =0.
h i h i h i The distances from some extremum and be-
These equations can be easily transformed to equations tw•een two extrema are defined as
for y . For example,
i
(φ,N)=
hφ1i†h∂V/∂φ†1i=y1T1+y3T3 =0, =(φ1 φ2 N φ2 φ1DN)†(φ1 φ2 N φ2 φ1 N )
h i − h i h i − h i i ≡
hφ2i†h∂V/∂φ†1i=y3∗T1+y2T3 =0, ≡x1y2+x2y1−x3y3∗ −x†3y3, (11)
hφ2i†h∂V/∂φ†2i=y2T2+y3∗T3∗ =0, D(I,II)≡hD(φ,II)iI≡hD(φ,I)iII≡D(II,I)=
φ1 † ∂V/∂φ†2 =y3T2+y1T3∗ =0. = ( φ1,I φ2,II φ2,I φ1,II ) 2.
h i h i | h ih i−h ih i |
4
Note that (φ,N) = 0 at φ = φ and (I,II) > 0 quantities y in a following way
i i N i
D h i D
for a pair of different extrema.
• The extremum energy is mm212212 ==22((λλ13yy11++λλ32yy22++λλ67yy33++λλ∗6∗7yy33∗∗ −yy21R)) ,, (14b)
ENext =V(hφiiN)≡ (12) m212 =2(λ6y1+λ7y2+λ5y3+λ4y3∗ −−y3∗RR) .
V(yi,N)=V0 V2(yi,N)+V4(yi,N).
≡ − The differentiation of (14a) gives for T :
i
According to theorem on homogeneous functions, in
each extremum point T1 =y2 , T2 =y1 , T3 = y3∗ , T3∗ = y3 .
R R − R − R
Λ y y =M y , or equivalently
ij i,N j,N i i,N In accordance with (8) for charged extremum we have
V2( φi N)=2V4( φi N) (13) from here =0.
h i h i ⇒ R
Fortheneutral extremumthe Higgsfieldsmassmatrix
⇒ENext =V0−V4(yi,N)=V0−V2(yi,N)/2. (10b) for the upper ( ) components can be written as
±
The extremum with the lowest value of energy (the
golfotbhaelmmoindieml.uOmthoefrpeoxtternetmiaal)carenabliezeesitthheervsaacdudulempsotianttes M++ = TT31∗ TT32 ≡ yy23 −yy13∗ R.
(cid:18) (cid:19) (cid:18)− (cid:19)
or maxima or local minima of the potential. It can be
established, in particular, by the study of the effective At Z =0 determinant of this matrix equals to 0. There-
mass matrix at these extrema. fore, one eigenstate of this matrix equals to 0. It de-
Decomposition around EWSB extremum. scribesmasslesscombinationofchargedHiggsfields(well
•
Our potential can be rewritten as a sum of extremum knownGoldstone state). The secondeigenstate of above
energy and terms vanishing in the extremum point to- matrix describes the physical charged Higgs boson with
getherwiththeirderivativesinφ . Thispolynomialwith mass
i
terms up to the second order in x can be written as a
i
sum of polynomials of second and first orders in xi van- MH2± =TrM++ =T1+T2 =(y1+y2)R.
ishing together with their derivatives in the extremum This quantity is positive for the minimum of the poten-
point. The form of second order polynomial is fixed by tial, it can be negative in other extremes. Finally, we
a quartic terms of potential, it can be only V4(xi yi). obtain
−
The residuary first order polynomial in x must be pro-
i
portional to (φ,N). Therefore =0 for charged extremum
D R
M2 (14c)
V =ENext+V4(xi−yi,N)+R·D(φ,N). (14a) R= y1+H±y2(cid:12)N for neutral extremumN.
(cid:12)
It means that the mass terms of (3) can be written via (cid:12)
(cid:12)
IV. EW SYMMETRY CONSERVING (EWC) POINT
The EWc point hφ1i=hφ2i=0 is extremum of potential. Depending on m2ij it has different nature:
At mm2121 <>00,,mm2222 <>00, and m211m222 ≥|m212|2 −mmainxiimmuumm,, (15)
11 22 −
At m2 m2 < m2 2 saddle point.
11 22 12
| | −
According to [10] no other extremum can be a maximum of potential.
V. CHARGED EXTREMUM withfermionswillnotpreserveelectriccharge,photonbe-
comes massive, etc. [7]. That is the reason why this ex-
We consider now the extremum which appears at tremum is called the charged extremum. We label quan-
tities related to this extremum by a subscript N =ch, if
Z =y1y2 y3∗y3 =0 u=0. (16) necessary.
− 6 ⇒ 6
If this extremum realizes the vacuum, it is not possi- Certainly, this case is not realized in our World. Nev-
ble to split the gauge boson mass matrix into the neu- ertheless,itisinterestingtoconsidermainfeaturesofthe
tral and charged sectors, the interaction of gauge bosons case when this extremum is the vacuum state in respect
5
totheopportunityofdifferentscenariosintheEarlyUni- The specific form of condition y1,2 > 0 depends on
verse. the value of parameter λ3 admissible by the positivity
Intheconsideredcaseeqs. (8)fortheextremumofthe constraint (4b):
potential have form
m211λ2>m222λ3, λ3
TTT213 ===λλλ421yyy321∗+++λλλ335yyy123+++λλλ∗7∗66yyy33∗∗1+++λλλ776yyy233−−mmm212221221///222===000,,, (17) mm212212λλ21<>mm222121λλ33, at √|λλ13λ|2 <1; (21b)
T3∗ =λ4y3+λ∗5y3∗+λ∗6y1+λ∗7y2−−m∗122/2=0. m222λ1<m211λ3 at √λ1λ2 >1.
onTlyhoanteisuanisqyusetesmoluotfiloinne(aerxecqeputatsioomnsefdoregyein.eIrtactaencahsaevse) If λ5 and m212 arereal(explicitly CP conservedpoten-
tial),condition(21a)forbidsalsosmallvaluesofλ4+λ5.
– system can have only one charged extremum.
The extremum energy in this case is subdivided into
Thissystemhassolutionatarbitraryparametersofthe
the sum of Z2 symmetry conserving term and Z2 sym-
potential. However, in accordance with (9), it describes
metry violating term:
anextremumoftheoriginalpotential(3)(wedefinesome
iqnueaqnutaitliiteisesviya1n>d0u,)yi2f o>nl0y,tZhe>o0bt(a6i)n.edInvtahluisescays1e,2 obey Ecehxt=−m411λ2+8m(λ4212λλ21−2λm23)211m222λ3 −
− (22)
v1 =√2y1, v2 = 2y3y3∗/y1, 2m†122m212λ4−m†142λ5−m412λ†5 .
u= 2Z/y|1,|e2iξp=y3/y3∗. (18) − 8(λ24−λ5λ†5)
This extremum is the minimum of the potential
p
Inequalities (6) determine the range of possible values of (charged minimum) if condition (19) is satisfied. In the
λi and m2ij where the charged extremum can exist. considered case it is easy to obtain (with argumentation
Condition for minimum. In the discussed case similar to that in [12]) that this condition is satisfied if
•
the potential (3) can be rewritten in the form (14) with
= 0. The charged extremum is minimum of the po- λ1 >0, λ2 >0, λ1λ2+λ3 >0, λ4 > λ5 .
R | |
tential if the quadratic form V4(xi yi,ch) is positively
defined at each classicalvalue of ope−rators x . This con- Note that latter conditpion guarantees negativity of sec-
i
dition differs from the positivity constraint (4a), since ond item in (22).
quantities z = x y do not need to satisfy condi-
i i i,ch
−
tions given in this constraint. Here z1 and z2 are some
real quantities (positive or negative) and z3 = z3∗∗ is an VI. NEUTRAL EXTREMA, GENERAL CASE
independent complex quantity. Therefore, the condition
for the charged minimum is (see e.g. [10]) Other solutions of the extremum condition (5) obey a
condition for U(1) symmetry of electromagnetism, that
V4(zi) 0atarbitrary realz1, z2, complex z3. (19) is solution with
≥
The case of softly broken Z2 symmetry Z =y1y2 y3∗y3 =0
• − ⇒
(λ6 = λ7 = 0). The main features of the solution 1 0 1 0
are seen in the case of the soft Z2 symmetry violation. ⇒hφ1i=√2 v1 , hφ2i=√2 v2 = v2 eiξ ,
In this case solution of equation (17) has form (cid:18) (cid:19) (cid:18) | | (cid:19)
another parameterization: (23)
y1 = m221(1λλ12λy−23−m=22λ2−23λ)m23(,21λ224λy2∗5+=λm5−λ∗1∗5222m)λ(21λ411.λλ32+−mλ22232)λ1 , (20) v2 =2(yv11+=yv2)ceo2=siξβ2=,(cid:0)|yhφ3v/12yi=|3∗2.+vs|ihnφβ2i,|2(cid:1)>0,
−
Conditions(6)limitthedomaininthespaceofparam- In this case quantities yi are not independent. There-
eters, where the charged extremum can exist. fore, the field values at the extremum point cannot be
The first condition in (6) reads as obtained by minimization of form (12) in yi. In these
terms system of equations for v.e.v.’s has form (8) with
(m211λ2 m222λ3)(m222λ1 m211λ3) solutionsyi =yi,n. Itisimportanttonotethatthenum-
− (λ1λ2 λ23)2 − − ber of independent parameters here is 3 (not 4). Those
m†122λ4−−m212λ†5 2 >0. (21a) aorfevtahleureesaloqfufianetlidtsiesayt1,thy2eaenxdtrtehme puhmaspeodiniffteξre(nncoet
−(cid:12)(cid:12) λ24−λ5λ†5 (cid:12)(cid:12) separate phases of these values!).
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
6
Charged Higgs mass. It is useful to reproduce parameters of the potential become complex, giving CP
•
here the equation for the charged Higgs mass, given e.g. violation in the Higgs sector (mixing of scalar and pseu-
in [4] (eq. (4.3)). doscalar neutral Higgs bosons) – see for details e.g. [4].
Thatisthereasonwhythisextremumiscalledthespon-
M2 =v2 Re m212−2(λ6y1+λ7y2) e−iξ taneously CP violating (sCPv) extremum [1, 7].
H± ( (cid:0)(cid:2) 4√y1y2 (cid:3) (cid:1)− (24a) The substitution of cosξ (26) into (25) results in
−λ4+Re(2λ5e−2iξ) . EsCPv = λ1y12+λ2y22 +λ345y1y2
(cid:27) 2 −
In addition, we present corrected eq. (4.5c) from [4] m211y1+m222y2 [m212 2(λ6y1+λ7y2)]2 (27)
− e
whichgivesmassofeitherpseudoscalarHiggsinthecase − 2 − 8λ5
of CP conservationor intermediate quantity obtained at
(here λ345 =λ3+λ4 λ5).
partial diagonalization of mass matrix in the case of CP −
violation Further minimization gives the system of linear equa-
e
v2 tions ∂EsCPv/∂y1 = 0, ∂EsCPv/∂y2 = 0 (we don’t
M33 ≡MA2 =MH2± + 2 λ4−Re(λ5e−2iξ) . (24b) present this general system only due to its bulkiness).
Only one solution of this system exists, so y1, y2 and
For the Higgs potential(cid:0)of general form w(cid:1)e have no cosξ are describedby parameters ofthe potential
•
idea about classification of neutral extrema. However, unambiguously. (This conclusioncanbe obtainedalso
if CP conservingextremum (with no scalar-pseudoscalar from the description of [10].)
mixing)exists,there isa basisin(φ1, φ2)spaceinwhich The discussed extremum can be realized only in the
potential has explicitly CP conservingform [5], [4] (with rangeof parametersof the potentialobeying inequalities
all real λ , m2). Using such a form of potential, the
i ij
subsequentusefulclassificationcanbeintroduced,inthis m212−2(λ6y1+λ7y2) = cosξ <1,
important case. (cid:12) 4λ5√y1y2 (cid:12) | | (28)
(cid:12) (cid:12)
(cid:12) y1 >0, y2 >(cid:12) 0.
(cid:12) (cid:12)
EXPVLIICI.ITNCEPUCTORNALSEERXVTARTEIOMNA(,RCEAASLEλOi,Fm2ij) theThexetcrehmanugme heφn2eirg→y (h2φ52)i.∗T(hξe→refo−reξ) does not modify
In accordance with definitions (9), we have for each if φ1 = φ1 , φ2 = φ2 is the extremum
solution y3 = √y1y2eiξ. In the discussed case the ex- of potentiahl i φ1 = φh1 ,iφ2 = φ2 ∗ is also
tremum energy (12) is transformed to the form the extremum⇒; these htwoi extremh aiare de- (29)
1 generate in energy[1]anddefinetwo”direc-
Eext =−2 m211y1+m222y2+2m212√y1y2cosξ + tions” ofCP violation,”left” and”right”,with
+λ5y+1yλ22c1(cid:8)oys122+ξ+λ222y(22λ6+y1(λ+3λ+7yλ24))y√1yy12y+2cosξ(cid:9). (25) ξ = arccos m212−2(λ6y1+λ7y2) . (30)
± 4λ5√y1y2
(cid:18) (cid:19)
Nowwefindextremaincoordinatesy1,y2,ξ. Westart
fromtheminimizationinξ atfixedy . Itgivestwotypes Our potential (25) is a second order polynomial in
i •
of solutions: cosξ. ThesCPvextremum(ifitexist)canbeaminimum
only if λ5 >0, in accordance with [6].
Substituting the cosξ (26) into (24a) we obtain al-
[A]: cosξ = m212−2(λ6y1+λ7y2), ter•native form for the mass of H± and M33 in sCPv
4λ5√y1y2 (26) extremum3
[B]: sinξ =0. v2
For the discussed explicitly CP conserving potential this MH2±,sCPv = 2 (λ5−λ4), (31)
equation is equivalent to the constraint eq. (3.11) ob- M33,sCPv =v2λ5sin2ξ.
tained in [4].
To realizeminimum of potential, allmass squaredeigen-
values and all diagonal terms of squared mass matrix
A. Spontaneously CP violating extremum
The extremum point (26[A]) describes a solution with
3 The same equation for the mass of H± was found in [7] and (for
complexvalueoffieldv2 atrealparametersofthepoten- thecaseofexactZ2 symmetry)in[14]. Notethatinthediscussion
tial. The rephasing transformation φ2 → φ2e−iξ trans- of sCPv extremum authors of [15]used equation forMA2, which is
forms the potential to the real vacuum form, in which incorrect inthiscase.
7
must be positive. Therefore,necessary conditions for re- extremum (28) can be written in the form
alization of sCPv minimum are
m211λ2 >λ345m222
λ5 >0, λ5 >λ4. (32) m222λ1 >λe345m211 ⇒
Note that the mass matrix for neutral Higgses in this ⇒ mm212212 >rλλ21eat√λ1λ2 >|λ345|,
scPv extremum can be written as (sign · · · mean the m211λ2 <λ345m222 e (35)
diagonalsymmetricquantityfromtheupperrightcorner
of matrix) m222λ1 <λe345m211 ⇒
M11 M12 v2(λ5sβcosξ+λ6cβ)sinξ ⇒ mm212212 <rλλ21eat√λ1λ2 <λ345,
M=··· M22 v2(λ5cβcλo5sξsin+2λξ7sβ)sinξ . m412 <16λ25y1y2. e
··· ···
 
It shows that – as it is naturally expected – the CP-
B. CP conserving extrema
violating field mixing is weak if the phase ξ is small.
Moreover, in the case of soft Z2 symmetry violation
The solution (26[B]) describes extrema that corre-
(λ6 =λ7 =0)physicalCPviolationis absentatξ =π/2
spond to ξ = 0 and ξ = π. The case ξ = π can be
(that corresponds to the exact Z2 symmetric case, with
m212 =0, λ6 =λ7 =0)4. toibvtea,ini.eed. afrlloomwtthaencβasteoξbe=n0egiaftwivee.alTlohwerve2fotroe,bweitnheoguat-
a. The case of softly broken Z2 symmetry (λ6 = 0, loss of generality we consider below the only case with
λ7 = 0). The analysis becomes more transparent for ξ =0.
the softly Z2 violating case at real λ5, m212. The cor- InthesecasesCPviolationdoesnotappear(CP con-
responding linear system is λ1y1 + λ345y2 = m211/2, serving – CPc – extrema). The extremum condition
λ2y2+λ345y1 =m222/2 with solution (8), writtenfor vi =√2yi, has formofthe systemoftwo
e cubic equations:
e
y1=m211λ22−∆λs345m222, y2=m222λ12−∆λs345m211 ⇒ m211v1++m(3212λv62v12=+λλ1v71v322+)vλ23,45v1v22+
taen2β = m222λ1−λ345m21e1 , m222v2+m212v1 =λ2v23+λ345v12v2+ (36)
⇒where ∆ms211=λ2λ1−λλ2ee3−45λm2342252; (33) λ+34(5λ6=v12λ+3+3λλ74v2+2)vλ15,.
m2
cosξ = 12 .e
4λ5√y1y2 Rewriting this system with parametrization
v1 = vcosβ, v2 = vsinβ, we express the quantity
v2 via t tanβ and obtain the equation for t, similar to
Inthis casethe extremumenergyis obtainedeasily from ≡
equations presented in [8]:
(12) (cf. eq. (22)):
m2 +tm2
v2 =(1+t2) 11 12
EsCPv =−m411λ2+m422λ81∆−s2m211m222λ345−m8λ4152 . (34) (1+t2)λ1+λ3t4m5t2222++3mλ2162t+λ7t3 ≡ (37a)
e ≡ λ2t3+λ345t+λ6+3λ7t2
and
Taking into accountthe positivity constraint(4b), the
necessary conditions for realization of the CP violating
(λ2m212 λ7m222)t4+
−
+(λ2m211+2λ7m212 λ345m222)t3+
−
4 Inthe exactZ2 symmetriccasesignofλ5 canbechanged bysim- +3(λ7m211−λ6m222)t2+ (37b)
plerephasingwithoutchangeofotherparametersofpotential. To- +(λ345m211 2λ6m212 λ1m222)t+
getherwithcondition (32)itmeansthatformalsCPvextremumin − −
thiscasegivenoCPviolation. +(λ6m211 λ1m212)=0.
−
8
It is easy to obtain that in this parametrization definition (11). Therefore by substraction of one equa-
tion from another we obtain5
v2 v2t2
v12 ≡2y1 = 1+t2, v22 ≡2y2 = 1+t2, (37c) ∆E(II,I)≡EIeIxt−EIext = 21(RI −RII)·D(I,II) (40)
v2t
v1v2 ≡2y3 =2y3∗ = 1+t2 . with = 0 for charged extremum and = MH±/v2 N
R R |
for neutral extrema (14). Therefore, the quantity
Nowonecanrewriteextremumenergy(13)fordiscussed ( I II) determines the hierarchy of extrema.
R −R
extremum in the form EWc (hφ1i = hφ2i = 0) and EWSB extrema.
•
Rewriting eq. (14) for any EWSB extremum one can
m2 +tm2 m2 +2tm2 +t2m2 obtain
11 12 11 12 22
= . (38)
ECPc −(cid:0) 8(λ1+λ(cid:1)34(cid:0)5t2+3λ6t+λ7t3) (cid:1) 0= EWc = EWSB +V4(yi,EWSB yi,EWc)
E E −
By construction, one should consider only real solu- with y =0
i,EWc
tions of equation (37b) satisfying v2 > 0. The equation ⇒
of fourth degree might have 0, 2 or 4 real solutions (in- EWSB = V4(yi,EWSB)<0.
E −
cludingaccidentaldegeneracy). Takingintoaccountpos-
Therefore,
sible negative values of v2 given by (37a) one can state
1. IfEWSBextremumisaminimumofpotential,then
carefully that there could be up to 4 CPc extrema.
the EWc extremum has higher energy.
Ifnecessary,welabeldifferentextremaofsuchtype with
an additional subscript N =I, II, .... 2. vIfatchueuEmWstcaetext(rietmcaunmh(ahpφp1ein=ohnφly2ia=tm02)r,emali2zes<th0e)
• Note that at m211 = m222, λ1 = λ2, λ6 = λ7 our po- allEWSBextremaareeithersaddlepo1i1nts o2r2local
tential has additional symmetry φ1 ↔φ2. The potential maxima of potential.
has extrema keeping this very symmetry (t = 1) and
± Neutral extremum and charged extremum.
thosewherethissymmetryisspontaneouslybroken. The •
1. According to eq. (14), the Higgs potential can be
latter states are degenerated in energy like sCPv states
written as a sum of chargedextremum energy ext
discussedinsect.VIIA. Ifthesestatesformminimumof Ech
potential,wedealwithtwodegeneratedminima(cf.[10]). and operator V4(xi −yi,ch). The latter is a poly-
nomial of the second degree in z = x y . If
Weak violation of mentioned symmetry destroys dis- i i − i,ch
charged extremum is a minimum of the potential,
cusseddegeneracy. Therefore,inthegeneralcaseinsome
region of parameters our system can have two CPc min- the quantity V4(zi) is positively definite at arbi-
ima simultaneously. trary real z1,2 and complex z3 (19). Since V4 is
quadratic form in z , this quantity is positive also
i
• In the case of soft Z2 symmetry violation in the points, correspondent all other extrema of
(λ6 =λ7 =0) equations (37) become: potential. Therefore, if charged extremum is
a minimum of the potential, it is the global
v2 =(1+t2)m211+tm212 (39a) minimum – the vacuum state6.
λ345t2+λ1 2. Besides,thedifferenceoftheextremumenergiesfor
neutral and charged extrema (40) can be written
as ∆ (ch,n) = (M2 /v2) (ch,n) (see also [7],
λ2m212t4+(λ2m211−λ345m222)t3+ (39b) [10]).ETherefore: H±,n n D
+(λ345m211 λ1m222)t λ1m212 =0. 2a. If the neutral extremum is a minimum of po-
− − tential, i.e. M2 >0, the charged extremum has
H±,n
a higher energy [7].
2b. If the charged extremum realizes the mini-
VIII. VACUUM AND OTHER EXTREMA
mum of the potential, all neutral extrema are sad-
dle points or local maximums (since it can take
Inthissectionwecalculatethe differenceofextremum
place only at M2 <0 for all n).
energies for different extrema. For this purpose we ex- H±,n
Two neutral extrema I and II.
press extremum energy in the extremum II via parame- •
For two neutral extrema the difference of extremum
ters of extremum I and vice versa with (14):
energies (40) is
EIeIxt =EIext+V4(yi,II −yi,I)+RI ·D(II,I), ∆ (II,I)= M2 /v2 M2 /v2 (II,I). (41)
EIext =EIeIxt+V4(yi,I −yi,II)+RII ·D(I,II). E (cid:16) H±,I I − H±,II II(cid:17)·D
The quadratic polynomial V4(zi) = V4( zi) (simultane-
−
ous change of signs of its arguments). The distance be- 5 Inparticularcasesthesamekindofequations wasobtained in[7].
tweentwoextrema (I,II)issymmetricandpositiveby 6 Thisconclusion canbeobtainedalsofromdiscussionofref.[13].
D
9
Therefore, in particular, a efficient form of the potential, determined by the form
1. If extremum I is minimum (M2 > 0) and ex- of the Yukawa sector (see discussion after (3)).
H±,I
tremum II is not a minimum with M2 < 0, There are two very different types of extremum of the
H±,II potential in 2HDM (9) – the charged extremum with
extremum II is higher than extremum I.
u = 0 (it does not describe the modern reality) and the
If two minima of the potential exist with energies 6
neutral extremum with u=0.
and , the energy interval ( , ) cannot
I II I II
E E E E A. General case.
contain saddle points or maxima.
1) The Electroweak symmetry conserving extremum
2. Fortwo neutralminima of potentialor a minimum
and a saddle point with MH2±,N > 0, the deeper m(hϕ2111,im=222 h<ϕ20i, =m2102)2c<anmr21e1amli22z2e.s Itnhethviasccuausme aslltaotteheirf
(a candidate for the global minimum – the vac- | |
extrema are saddle points.
uum)istheextremumwiththelargervalueofratio
2) The charged extremum is determined by parame-
M2 /v2 .
H±,N N ters of the potential uniquely by the system of linear
Explicitly CP conserving potential. Neutral equations (17), (18). It exists if solutions of this system
•
extrema. obey conditions (16).
1. sCPv and CPc extrema. If the charged extremum realizes the minimum of the
We have MH2±/v2|sCPv = (λ5 −λ4)/2 (31) . In potential, itdescribes the globalminimum –the vacuum
accordance with (24b) we have for CPc extremum state. In this case all neutral EWSB extrema are saddle
MH2±/v2|CPc−(λ5−λ4)/2=MA2/v2|CPc. Itallows points of the potential.
us to rewrite eq. (41) in the form (see [7]) 3) The number of neutral extrema is more than one
(in addition to EW symmetry conserving extremum
∆ =M2/v2 (sCPv,CPc) (42)
EsCPv,CPc A |CPc·D φ1 = φ2 =0).
h i h i
The potentialcanhavesimultaneouslytwoneutralex-
with positive (sCPv,CPc).
D trema I and II. If M2 /v2 M2 /v2 >0, the state
Therefore, similarly to the comparison of charged H±,I I − H±,II II
I is below state II (state II cannot be vacuum) (41).
and neutral extrema:
B. Explicitly CP conserving potential (with all
1a) If system has a sCPv minimum, i. e.
M2/v2 < 0, the minimum is the vacuum. All real coefficients in (3)). In this case one can distinguish
A |CPc aCPconserving(CPc)extremumwithzerophasediffer-
CPc extrema are saddle points, not minima.
ence between the values φ at the extremum point and
1b) If system has a CPc minimum i. e. h ii
spontaneously CP violating (sCPv) extrema, in which
M2/v2 > 0, the sCPv extremum cannot be
a mAinim|CuPmc, it is a saddle point. the phase difference between the values hφii is nonzero,
the lattergeneratesneutralHiggsstateswithout definite
Thetoymodelgivesgoodillustrationforthisstate-
CP parity. Total number of extrema in this case can be
ment (see Fig. 2).
upto 8 (0or 1chargedextremum, upto 4 CPcextrema,
2. Two CPc extrema I and II in the case of the
2 or 0 sCPv extrema, 1 EWc extremum).
softly Z2 violating potential. 1) The sCPv extremum is determined by the parame-
In this case one can use the eq. (24a)
ters of the potential uniquely by system of linear equa-
2MH2±/v2 = (m212/v1v2) − λ4 − λ5. It allows tions. Itexistsifsolutionsofthis systemobeythe condi-
to transform the mass factor in (41) to the
II,I tion (28). The sCPv extremum state is doubly degener-
M
form
ateinsignofphasedifferencebetweenthevaluesoffields
m2 m2 at the extremum point.
II,I = 12 12 (43) For this extremum to be minimum it is necessary to
M (cid:18)2v1,Iv2,I − 2v1,IIv2,II(cid:19) have λ5 > 0 and λ5 > λ4. If sCPv extremum realize
or – with the aid of (37) – to express via minimumofpotential,itisthevacuumstate(doublyde-
II,I
solutions of equation for tanβ =t. M generated). In this case all EWSB extrema are saddle
So that, in this case the hierarchy of extrema can points. (In MSSM with loop correction λ5 < 0, there-
be established without direct calculation of M2 . fore the sCPv extremum cannot be the vacuum (see e.g.
H±
[15]).)
2) System can have more than one CPc local minima,
IX. GENERAL PICTURE the vacuum state is the lowest among them. For the
important case of softly broken Z2 symmetry, if in this
case we have two CPc minima of the potential, I and
Let us summarize the picture obtained.
II,the eq. (43) means that the state I is below state II
In our discussion of a separate extremum we have in
and can describe vacuum if
mindsuchachoiceofthez axisintheweakisospinspace
that in this extremumhφ1i= v01 with realv1 >0 (9). m212/(vI2sin2βI)−m212/(vI2Isin2βII)>0. (44)
(cid:18) (cid:19)
We think that it is unnecessary to search for the basic C. The decomposition of potential near extrema (14)
independent form of the results. In our opinion there is seems to be useful for phenomenological analysis.
10
D. For explicitly CP conserving potential we have tion of the Gibbs potential become invalid (evolution of
foundexplicitequationsforextremumenergies(22),(34), Universe can be faster than transition to the thermody-
(38)viaparametersofpotentialandsetofnecessarycon- namical equilibrium).
ditionsforrealizationofchargedorsCPvextrema. These 2) If during evolution, Universe changes different
equations allow to pick out extremum with lower energy phases of 2HDM, these phase transitions look as tran-
– vacuum state and to look for phase transitions at the sitions of the second order as long as we consider effects
variation of parameters of potential. in the tree approximation (fluctuations = multiloop ef-
fectscanchangetypeoftransition). Ineachphaseduring
cooling the masses of the particles of matter (which are
X. WHAT NEXT given by values φ1 and φ2 ) evolve and even their hi-
h i h i
erarchy can change. At phase transition the speed of
Nowwediscussbrieflypossibleeffectsofradiativecor- variationof these φ1 and φ2 and even their interrela-
h i h i
rections and present first ideas about and the impact of tion are changed. Some examples of this type gives toy
our analysis on cosmology. These are the problems for model, considered in Appendix.
future studies. Eachof these transitions is accompaniedby formation
Radiative corrections. With radiative (loop) cor- of bubbles of the old phase within the new one. Since
•
rections main qualitative features of obtained picture cooling of Universe is very fast, these bubbles can be
will be changed weakly (provided these corrections are frozen in the new phase for a relatively long time (like
small7). These corrections are important if they vio- in supercooled vapor). We can observe now some effects
late some artificial symmetry of the potential. In our fromdifferentseriesofthesebubblesdespitethefactthat
case that is explicitly CP conserving form of poten- these bubbles have disappeared by today.
tial. Radiative corrections contain contributions e.g. 3) It is very attractive to assume that this very al-
of light quarks, having imaginary parts for the consid- most degenerate sCPv state is realized now, ex-
ered mass interval. (The simplest example gives cor- plaining observed CP violation. In this case at the first
rection to λ5 term, obliged by interaction with b-quark. stage during the evolution of Universe domains of both
Very rough estimate of loop correction gives additional types of the CP violationwould form. Then, with mech-
Imλ5 .(mb/v)4(mb/Mh)2 10−10, where factors mb/v anisms like those discussed in [18] these domains decay
are from Yukawa coupling a∼nd factor (m /M )2 – from to the phase with left violationofCP symmetry. In con-
b h
loopintegralitself.) Theseimaginarypartseliminate de- trast to the simple case considered in [18] domain wall
generacyofthesCPvextremain accordance with the can be high enough due to complex structure of poten-
arrow of time, and it is natural to expect that the en- tial and the presence of a number of particles of matter
ergy difference between these two states is small – we within the walls. An important feature of the walls is
deal with almost degenerate states. In simple words, one that the matter within wall is much heavier than in vac-
can write that the phase with left violation of CP is real uum. Moreover, the profile of this wall in φ space can
i
vacuum. becomplicatedsothatwithinthewallthephotoncanbe
Possible effects for cosmology. massive (like in charged extremum) or not (if the saddle
•
1) Temperature dependence of the Gibbs po- point between the minima obeys U(1) symmetry of elec-
tential is determined by standard methods of statisti- tromagnetism). In the former case the domain walls are
cal physics (see e.g. [16]). In the first approximation of opaque. In the second case these walls can be gray due
perturbation theory only mass term modifies. At large to fluctuations.
enough temperature T we have ∆m2 = a T2(φ†φ )/2. The comparison of speed of elimination of domains
ij ij i j
Higher order corrections modify these a and change with the rate of cooling of Universe looks an interesting
ij
weakly (no more than logaritmically) parameters λ . problem.
i
Therefore general features of phase transitions during In the considered case a small difference between the
evolution of Universe can be analyzed in terms of vari- energies of two sCPv phases can have relation to the
ation of mass term of potential (3) with the aid of eq-s value of the cosmologicalconstant.
(22), (34), (38) (see [17]). This picture can be modified
by variation of form of Gibbs potential near phase tran- Acknowledgments. We are thankful I. Ivanov,
sition. M. Krawczyk,L. Okun, R. Santos, A. Slavnov for useful
Moreover, in vicinity of phase transition all processes discussions. This research has been supported by Rus-
become slower,and adiabatic approximationfor calcula- sian grants RFBR 05-02-16211,NSh-5362.2006.2.
7 In the discussion of Higgs sector in MSSM 1-loop corrections are
notsmall.
[1] T. D. Lee. Phys. Rev. D 81226 (1973) [2] J.F.Gunion,H.E.Haber,G.Kane,S.Dawson,TheHiggs